Finite Pressure Corrections the to Parton Structure of Baryon Inside a Nuclear Medium.
Our model calculations performed in the frame of the Relativistic Mean Field (RMF) approach show how important are the modifications of baryon Structure Function (SF) in Nuclear Matter (NM) above the saturation point. They originated from the conservation of a parton longitudinal momenta - essential in the explanation of the EMC effect at the saturation point of NM. For higher density the finite pressure corrections emerge from the Hugenholtz -van Hove theorem valid for NM. The density evolution of the nuclear SF seems to be stronger for higher densities. Here we show that the course of Equation o State (EoS) in our modified Walecka model is very close to that obtained from extensive DBHF calculations with a Bonn A potential. The nuclear compressibility decreases. Our model - a nonlinear extension of nuclear RMF, has no additional parameters. Modelling deep inelastic scattering on nuclear, neutron or (strange) matter with finite pressure, we attempt to predict also the change of baryon masses in a strange nuclear medium. These changes are derived from the Momentum Sum Rule (MSR) of quark longitudinal momenta for different constituents. The increasing pressure between baryons starts to increase baryon Fermi energies in comparison to average baryon energies , and consequently the MSR is broken by the factor from the hadron level in the convolution model. To compensate this factor which increases the longitudinal momentum for nuclear partons, the baryon SF in the nuclear medium and their masses have to be adjusted. Here we assume that, independently from density, quarks and gluons carry the same amount of longitudinal momenta - the similar assumption is used in the most nuclear models with parton degrees of freedom.
I The nuclear deeply inelastic limit - nuclear equilibrium
In the nuclear deep inelastic scattering on nuclei our time-distance resolution is given by variable zJaffe ():
which measures the propagation time of the hit quark caring the fraction of the longitudinal momentum of the nucleon of mass . Start with the scenario where the partonic mean free paths are much shorter then the average distances between nucleons. This means that partons (inside mesons) remain inside the ”volume” of a given nucleon and therefore we can treat nucleons as noninteracting objects remaining on the energy shell not affected by neighboring nucleons.
In the light cone formulationJaffe (); Fran (), corresponds to the nuclear fraction of quark longitudinal momentum and is equal (in the nuclear rest frame) to the ratio . But the composite nucleus is made of hadrons which are distributed with longitudinal momenta , where stands for nucleons, virtual pions, … . In the convolution modelJaffe (); Fran () a fraction of parton longitudinal momenta in the nucleus is given as the product of fractions: parton momenta in hadrons and longitudinal momenta of hadrons in the nucleus . The nuclear dynamics of given hadrons in the nucleus is described by the distribution function and SF describes its parton structure. In the convolution model restricted to nucleons and pions (lightest virtual mesons) the nuclear SF is described by the formula:
where and are the parton distributions in the virtual pion and in the bound nucleon. The nucleon distribution in the basic convolution formula can be simplified in the RMF to the formMike ():
Here the nucleon spectral function was taken in the impulse approximation: and . is the nucleon Fermi energy and takes the values given by the inequality .
The MSR for the nucleonic part is sensitive to the Fermi energy as can be seen from the integral:
Thus the nucleonic part of MSR gives a factor which is equal to at the saturation point. If the nucleon SF is not changed in the medium (no EMC effect except Fermi motion) the total (LABEL:SR) MSR is satisfied without nuclear pions:
Summarizing, good description jacek (); RW () of these deeply inelastic processes without gluon degrees of freedom allows us to assume that fraction of momentum carried by quarks does not change from nucleon to nucleus ( one half, the rest is carried by gluons). We will assume that balance also above the saturation point of NM. Now for the non zero pressure the Fermi energy in NM is no longer equal to the average binding energy, and corrections to MSR (4) proportional to the pressure which will now be investigated.
Ii Non-equilibrium correction to nuclear distribution.
For finite pressure very important is well known Hugenholtz van Hove relation between , and pressure (see for example) kumar (). The Fermi energy is defined as density derivative of the total nuclear energy :
where gives the volume. At the saturation point . But for negative pressure
and we have room for additional pion inside NM (even with unmodified nucleons) in the Björken limit (parton picture).
ii.1 Positive pressure.
Consider the additional pion contributions. The amount of of the total nuclear momentumjacek (); RW (); drell (); ms2 () was estimated from (LABEL:sum) for due to the smaller, dependent nucleon mass . For higher density, the average distances between nucleons are smaller, therefore parameter (LABEL:eq:mamam2) will increase with density. It means that the room for nuclear pions given by (LABEL:sum) from dependent nucleon mass will be reduced for higher densities and a dependence of from Björken will vanish gradually. Also the pion effective cross section is strongly reduced at high nuclear densities above the threshold in reaction calculated in Dirac-Brueckner approach hm1 () ( also with RPA insertions to self energy of and oset () included). Therefore for positive pressure the nuclear pions carry much less then of the nuclear longitudinal momentum and dealing with a non-equilibrium correction to the nuclear distribution (2) we will restrict considerations to the nucleon part without additional virtual pions between them
The Equation of State (EOS) for NM has to match the saturation point with compressibility but then the behavior for higher densities is different for different RMF models. We compare here two extreme examples: stiff model of Walecka wa () and nonrelativistic expansion in powers of Fermi momentum dabrnuc. For linear coupling in the standard Walecka model at saturation density of NM compressibility is too large ( MeV). The energy shown in Fig.LABEL:meson influences the nuclear EOS. In both models, two coupling constants of the theory are fixed by the empirical saturation density. Our approach is different. In this work we consider the change of the nucleon mass with the change of the parton distribution (nucleon SF) above the saturation point. The increasing pressure between nucleons starts to increase the (4) and consequently the sum rule (5) is broken by the factor . To compensate this factor which increases the longitudinal momentum (5) of nuclear partons, the nucleon SF in the nuclear medium has to be changed. For good estimate, in order to proceed without new parameters, assume (similarly to (LABEL:scale)) that the changes of SF will be included through the changes of Björken in the medium. Multiplying the argument of the SF by a factor the SF will be squeezed towards smaller and the total fraction of longitudinal momentum will be smaller by a factor :
Here in the integral we neglect the small contributions from region originated from NN correlations.
This means that quarks in the nucleus carry the same fraction of longitudinal momentum as in bare nucleons.
On the other hand the integral (9) corresponds (LABEL:eq) to the total sum of the quark longitudinal momenta inside a nucleon, which is proportional to a total nucleon rest energy or the nucleon mass. Consequently, the nucleon mass will be changed for to the mass by the gradually decreasing factor :
Here, our model applied to the linear Walecka model for nuclear and neutron matter make the EoS softer, close to semi-empirical analysispawel () and close to DBHF calculation with a realistic Bonn A potentialbonn (). Other features of the Walecka model, including a good value of the spin-orbit force remain in our model unchanged. Our results suggest corrections above the saturation density to any RMF model, with constant nucleon mass and unmodified parton SF. The stiffness of EoS recently discussedNSTARS () in application to compact and neutron stars is important when studying star properties (mass-radius constraint). Partial support of the Ministry of Science and Higher Education under the Research Project No. N N202046237 for is acknowledged.
- (1) R. L. Jaffe, Los Alamos School on Nuclear Physics, CTP 1261, Los Alamos, July 1985.
- (2) N. N. Nikolaev and V. I. Zakharov, Phys. Lett. B 55, 397 (1975). L. L. Frankfurt and M. I. Strikman, Nucl. Phys. B 316 (1989).
- (3) L. L. Frankfurt and M. I. Strikman, Phys. Rep. 160, 235 (1988).
- (4) S.V. Akulinichev, S. Shlomo, S.A. Kulagin,G.M. Vagradov, Phys. Rev. Lett. 55, 2239 (1985); G.V. Dunne and A.W. Thomas, Phys. Rev. D 33, 2061 (1986); Nucl.Phys. A 455, 701 (1986), R.P. Bickerstaff, M.C. Birse and G.A. Miller, Phys. Rev. D33, 3228 (1986); M. Birse, Phys. Lett. B 299, 188 (1993); K. Saito, A. W. Thomas, Nucl. Phys. A4745, 659 (1994); H.Mineo, W.Bentz, N.Ishii, A. W. Thomas and K.Yazaki, Nucl. Phys. A735, 482 (2004) J. R. Smith and G. A. Miller, Phys. Rev. Lett. 91 (2003) 212301.
- (5) J. R. Smith and G. A. Miller, Phys. Rev. C 65, 015211, 055206 (2002).
- (6) J. Rożynek, Nucl. Phys. A 755, 357c (2004).
- (7) J. Rożynek, G.Wilk, Phys. Rev. C 71, 068202 (2005).
- (8) M. Birse, Phys. Lett. B, 299, 188 (1993); L.L. Frankfurt and M.I. Strickman, Phys. Lett. 183B, 254 (1987).
- (9) D.M. Alde et al., Phys. Rev. Lett. 64, 2479 (1990).
- (10) K. Kumar, ”Perturbation Theory and the Many Body Problem”, North Holland, Amsterdam 1962.
- (11) B. ter Haar and R. Malfliet, Phys. Rev. C 36, 1611 (1987), Phys. Rep. 149, 287 (1987).
- (12) E. Oset, L.L. Salcedo, Nucl. Phys. 468, 631 (1987), ”The Nuclear Methods and the Nuclear Equation of State”, ed. M. Baldo, World Scientific 1999.
- (13) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. Vol. 16 (Plenum, N. Y. 1986).
- (14) J. Zimanyi and S.A. Moszkowski, Phys. Rev. C 42, 1416 (1990).
- (15) A. Delfino, C.T. Coelho and M. Malheiro, Phys. Rev. C 51, 2188 (1995), D. P. Menezes, C. Providencia, M. Chiapparini, M. E. Bracco, A. Delfino, M. Malheiro, Phys. Rev .C 76:064902, (2007).
- (16) N.K. Glendenning, F. Weber, S.A. Moszkowski, Phys. Rev. C 45, 844 (1992).
- (17) R. J. Furnstahl and B. D. Serot, Phys. Rev. C 41, 262 (1990).
- (18) P. Danielewicz, R. Lacey, W. G. Lynch, Science 298, 1592 (2002).
- (19) T. Gross-Boelting, C. Fuchs, A. Faessler, Nuclear Physics A 648, 105 (1999); E. N. E. van Dalen, C.Fuchs, A. Faessler, Phys. Rev. Lett. 95, 022302 (2005); Fuchs J. Phys. G 35, 014049 (2008).
- (20) T. Klähn, D. Blaschke, S. Typel, E. N. E. van Dalen, A. Faessler, C. Fuchs, T. Gaitanos, H. Grigorian, A. Ho, E. E. Kolomeitsev, M. C. Miller, G. Röpke, J. Trümper, D. N. Voskresensky, F. Weber and H. H. Wolter, Phys. Rev. C 74, 035802 (2006).